Can you be allergic to salt?

No. Your immune system can generate allergic antibodies only against large molecules such as proteins. Ions such as the sodium and chlorine ions in salt are too small to be able to trigger an antibody response.
No. No, salt exists as part of our normal body chemistry so you can't be allergic to it. However, depending on how the salt is processed, you might be sensitive (not allergic) to one of the chemicals used in production.